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A Simple Unsupervised Color Image Segmentation Method based on MRF-MAP
arXiv:1202.4237v1 [cs.CV] 20 Feb 2012
Qiyang Zhao
Abstract Color image segmentation is an important topic in the image processing field. MRF-MAP is often adopted in the unsupervised segmentation methods, but their performance are far behind recent interactive segmentation tools supervised by user inputs. Furthermore, the existing related unsupervised methods also suffer from the low efficiency, and high risk of being trapped in the local optima, because MRF-MAP is currently solved by iterative frameworks with inaccurate initial color distribution models. To address these problems, the letter designs an efficient method to calculate the energy functions approximately in the non-iteration style, and proposes a new binary segmentation algorithm based on the slightly tuned Lanczos eigensolver. The experiments demonstrate that the new algorithm achieves competitive performance compared with two state-of-art segmentation methods.
Index Terms Image segmentation, Markov random fields, maximum a posteriori, unsupervised segmentation.
I. I NTRODUCTION Unsupervised color image segmentation is important in various image processing and computer vison applications, such as medical imaging [1], image retrieval [2], image editing [3], and object recognition [4]. Estimating the maximum a posteriori (MAP) on the Markov random fields (MRF), is so far an fundamental tool which is widely adopted both in the unsupervised color image segmentations and supervised ones [5]-[10]. In all existing MRF-MAP-based image segmentation methods, their goals are to find the optimal label configurations on pixels to maximize the posterior probability which is in Qiyang Zhao is with the School of Computer Science and Engineering, Beihang University, Beijing, 100191, China. E-mail:
[email protected].
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proportion to the product of the MRF priors and the likelyhooods terms, or equivalently, to minimize the energy function of the smoothness terms plus data terms. The likelyhood/data terms consist of the parameters of some color distribution models, here these models specify the probabilities of any color occurring in each segmentation. These parameters usually derive from user interactions or random sampling, hereby there is always non-negligible inaccuracy in the likelyhood terms. In order to address this, the Expectation-Maximization (EM), simulated annealing and other iterative methods are usually adopted to progressively approach the appropriate parameters, especially in the case of unsupervised segmentations up to today [5]-[7]. There are many choices of optimization algorithms to be adopted in the M-step [11]. There are three major disadvantages in the current unsupervised segmentation methods. The first is the low efficiency of the iterative frameworks, particularly when faced with large size images. The second is always on the high risk to be trapped to the local optima of the energy function. Although stepping out and restarting the iteration is a reasonable improvement, there would inevitably be additional computational load and it is possible to be trapped again. The third is the coarseness in the segmentation results, and it is partially caused by the roughness of the likelyhood parameters. To address these issues, the letter proposes a new unsupervised binary segmentation method based on the approximation of the likelyhood terms, where the iterative computations are replaced by only one single step of solving the eigenvector of the largest eigenvalue, therefore the computational efficiency is remarkably improved. This new approach increases the chance to high quality segmentation results by obtaining the nearly optimal solutions to maximize the posterior probability as possible. It also provides us an effective way to test and verify the MRF prior parameters or their involved generating schemes, which are also critical to the segmentation tasks. II. C OLOR I MAGE S EGMENTATION
BASED ON
MRF-MAP
In the following sections, we focus on the binary segmentation with the label set {fore, back}. The computational goal of these methods is to maximize the probability P (L|I) of the segmentation label configuration L given an image I . According to the Bayesian rule, it is equivalent to maximize the joint probability P (I, L) = P (I|L) · P (L), where the prior P (L) is established on the Markov random field of I , the conditional probability P (I|L) is the likelyhood that the pixel colors occur in their corresponding
segments marked by different labels. In a more prevailing view, what we need is to minimize a energy function E which is the negative log-hood of P (I, L). Here E is usually written in the form of a data
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term ED plus a smoothness term ES multiplied by a factor λ: E = ED + λES
(1)
where ED reflects the likelyhood of the color occurrences in the image segments, ES is the sum of all adjacency interaction potentials of each two neighboring pixels of different labels: ED =
X
− ln PL(p) (p), ES =
p
X
S(c(p), c(q))
(2)
(p, q) ∈ N L(p) 6= L(q)
where L(·) is the pixel label, c(·) is the pixel color, and S(·, ·) is the perceptually similarity weight of two colors. It is meant two pixels p and q are adjacent to each other by noting (p, q) ∈ N . Thereafter, the segmentation task is to pursue an appropriate a label configuration to reach the lowest energy. Although as indispensable as the data terms when computing the energy functions, the smoothness terms are not to be addressed in the letter. There are two steps when determining the coefficients in the data term ED . First, a suitable color statistical model should be chosen. Histograms are usually adopted for images of small color spaces, such as gray scale or 256 colors, but it is not suitable for large color spaces as the samples were statistically too few when facing so many histogram bins. Some other models fit large color spaces well, and make an appropriate comprise between the efficiency and accuracy, such as the Gaussian mixture model (GMM). Second, the model parameters should be determined. However here arises a chicken or the egg dilemma unavoidably: we have to know the parameters first to minimize the energy function to obtain the optimal segmentation, but the optimal segmentation is just the key to produce the accurate parameters mentioned above. The usually adopted solutions to this are the iterated procedures, such as EM, in which the estimation and optimization are performed sequently but isolatedly in each single loop. Here the initial parameters are determined from sample pixels chosen by user interactions or random samplings. There are many choices to perform the optimization [11]: graph cut, Loopy Belief Propagation (LBP) and Iterated Conditional Model (ICM). The low computational efficiency is an adherent shortcoming of the iterated solutions, as it is extremely hard to predict when and where the iterations would halt. Furthermore, although the minimum of the energy function is an unambiguous target itself, the actual aim of the iterated solutions is not mathematically explicit for us to approach. As a result, the iterations are likely finished at the local minimal in most cases. To address these issues, the letter proposes an approximating expression which is rather close to ED in (2), and associate the approximated target energy function with the cut on a February 21, 2012
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complete graph G which has both positive and negative edge weights. In the following manipulations, it is rather straightforward to solve an eigen-system to pursue the minimum cut C on G, so to minimize the energy. There the expected segmentations are worked out directly without considering the troublesome parameters of the data terms at all. III. S EGMENTATION
BASED ON
A PPROXIMATE MRF-MAP
Consider there are n pixels of m(m
